Abstract
Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.
Highlights
A draft person’s physical spline, i.e., a thin elastic rod, has long been a model for smooth curves
By minimizing the Willmore energy in a xed conformal class, we provide a new numerical framework to experimentally study constrained Willmore surfaces of any genus
7 CONCLUSION We presented conformally constrained variational problems as a new framework for solving problems in mathematical visualization and geometry processing
Summary
A draft person’s physical spline, i.e., a thin elastic rod, has long been a model for smooth curves. Degeneration of meshes during optimization is otherwise a common problem [Bobenko and Schröder 2005] (see Figure 13) and has motivated methods based on conformal deformations [Crane et al 2011, 2013] or the addition of conformal penalty forces [Barrett et al 2016; Gruber and Aulisa 2020] Neither of these approaches though are entirely satisfactory since they exhibit numerical drift (Figure 12) in the conformal class resp. Aside from making our approach compatible with the many existing geometry processing algorithms which use dierential coordinates, it enables us to nd conformal immersions for abstract metric surfaces, valuable in mathematical visualization For triangle meshes such dierential coordinates are piecewise maps, constant per triangle, and we allow them to vary independently [Custers and Vaxman 2020]. Dierentials as variables allow us to interpret the Lagrange multipliers associated with the conformal class constraint as quadratic dierentials [Weber et al 2012] and, in a mechanical analogy, as stress tensors with corresponding forces acting along edges to counteract anisotropic distortion
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